SPRING 2019 NEW YORK UNIVERSITY SCHOOL OF LAW
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SPRING 2019
NEW YORK UNIVERSITY
SCHOOL OF LAW
“A theory of cooperation in games with
an application to market socialism”
AND
“Cooperation, altruism and economic theory”
John Roemer
Yale University
February 12, 2019
Vanderbilt Hall – 208
Time: 4:00 – 5:50 p.m.
Week 4SCHEDULE FOR 2019 NYU TAX POLICY COLLOQUIUM
(All sessions meet from 4:00-5:50 pm in Vanderbilt 208, NYU Law School)
1. Tuesday, January 22 – Stefanie Stantcheva, Harvard Economics Department.
2. Tuesday, January 29 – Rebecca Kysar, Fordham Law School.
3. Tuesday, February 5 – David Kamin, NYU Law School.
4. Tuesday, February 12 – John Roemer, Yale University Economics and Political
Science Departments.
5. Tuesday, February 19 – Susan Morse, University of Texas at Austin Law School.
6. Tuesday, February 26 – Ruud de Mooij, International Monetary Fund.
7. Tuesday, March 5 – Richard Reinhold, NYU School of Law.
8. Tuesday, March 12 – Tatiana Homonoff, NYU Wagner School.
9. Tuesday, March 26 – Jeffery Hoopes, UNC Kenan-Flagler Business School.
10. Tuesday, April 2 – Omri Marian, University of California at Irvine School of Law.
11. Tuesday, April 9 – Steven Bank, UCLA Law School.
12. Tuesday, April 16 – Dayanand Manoli, University of Texas at Austin Department of
Economics.
13. Tuesday, April 23 – Sara Sternberg Greene, Duke Law School.
14. Tuesday, April 30 – Wei Cui, University of British Columbia Law School.June 1, 2018. To appear in a symposium on this topic in the Review of Social Economy,
edited by Roberto Veneziani
“A theory of cooperation in games with an application to market socialism”
by
John E. Roemer**
Yale University
John.roemer@yale.edu
Abstract. Economic theory has focused almost exclusively on how humans compete with
each other in their economic activity, culminating in general equilibrium (Walras-Arrow-
Debreu) and game theory (Cournot-Nash). Cooperation in economic activity is, however,
important, and is virtually ignored. Because our models influence our view of the world,
this theoretical lacuna biases economists’ interpretation of economic behavior. Here, I
propose models that provide micro-foundations for how cooperation is decentralized by
economic agents. It is incorrect, in particular, to view competition as decentralized and
cooperation as organized only by central diktat. My approach is not to alter preferences,
which is the strategy behavioral economists have adopted to model cooperation, but
rather to alter the way that agents optimize. Whereas Nash optimizers view other players
in the game as part of the environment (parameters), Kantian optimizers view them as
part of action. When formalized, this approach resolves the two major failures of Nash
optimization from a welfare viewpoint -- the Pareto inefficiency of equilibria in
common-pool resource problems (the tragedy of the commons) and the inefficiency of
equilibria in public-good games (the free rider problem). An application to market
socialism shows that the problems of efficiency and distribution can be completely
separated: the dead-weight loss of taxation disappears.
Key words: Kantian equilibrium, cooperation, tragedy of the commons, free rider
problem, market socialism
JEL codes: D70, D50, D60, D70
**
I am grateful to Roberto Veneziani for organizing the symposium that led to this issue,
and to the authors of the contributions herein for stimulating my thinking on Kantian
cooperation.1
1. Man, the cooperative great ape
It has become commonplace to observe that, among the five species of great ape,
homo sapiens is by far the most cooperative. Fascinating experiments with infant
humans and chimpanzees, conducted by Michael Tomasello and others, give credence to
the claim that a cooperative protocol is wired into to the human brain, and not to the
chimpanzee brain. Tomasello’s work, summarized in two recent books with similar
titles (2014, 2016), grounds the explanation of humans’ ability to cooperate with each
other in their capacity to engage in joint intentionality, which is based upon a common
knowledge of purpose, and trust.
There are fascinating evolutionary indications of early cooperative behavior
among humans. I mention two: pointing and miming, and the sclera of the eye.
Pointing and miming are pre-linguistic forms of communicating, probably having
evolved due to their usefulness in cooperative pursuit of prey. If you and I were only
competitors, I would have no interest in indicating the appearance of an animal that we,
together, could catch and share. Similarly, the sclera (whites of the eyes) allow you to
see what I am gazing it: if we cooperate in hunting, it helps me that you can see the
animal I have spotted, for then we can trap it together and share it. Other great apes do
not point and mime, nor do they possess sclera.
Biologists have also argued that language would likely not have evolved in a non-
cooperative species (Dunbar[2009] ). If we were simply competitive, why should you
believe what I would tell you? Language, if it began to appear in a non-cooperative
species, would die out for lack of utility. The problem of cheap talk would be severe.
In addition, language is useful for coordinating complex activities – that is, ones that
require cooperation. It would not have been worth Nature’s investment in a linguistic
organ, were the species not already capable of cooperation, so the argument goes.
Cooperation must be distinguished from altruism. Altruism comes in three
varieties: biological, instrumental, and psychological. Biological altruism is a hard-
wired tendency to sacrifice for others of one’s species, which sometimes evolved through
standard natural selection, as with bees and termites. Some people speak of2
instrumental altruism, which is acting to improve the welfare of another, in expectation of
a reciprocation at some time in the future. It is questionable whether this should be
called altruism at all, rather than non-myopic self-interest. Psychological altruism is
caring about the welfare of others: it is a kind of preference. It is intentional, but not
motivated by self-interest, as instrumental altruism is. Psychological altruism is what
economists usually mean by the term.
Cooperation is not the same as psychological altruism. I may cooperate with you
in building a house because doing so is the only way I can provide myself with decent
shelter. It is of no particular importance to me that the house will also shelter you.
Cooperation is, I believe, a more generalized tendency in humans than altruism. One
typically feels altruism towards kin and close friends, but is willing to cooperate with a
much wider circle. With the goal of improving human society, I think it is much safer
to exploit our cooperative tendencies more fully, than our altruistic ones.
The examples I gave above of cooperation are quite primitive. Humans have, of
course, engaged in much more protracted and complex examples of cooperation than
hunting. We live in large cities, cheek by jowl, with a trivial amount of violence. We
live in large states, encompassing millions or hundreds of millions, in peace. Early
human society (in its hunter-gatherer phase) was characterized by peace in small groups,
up to perhaps several hundred, but by war between groups. Our great achievement has
been to extend the ambit of peaceful coexistence and cooperation to groups of hundreds
of millions, groups between which war continues to exist. In this sense, cooperation has
expanded immeasurably since early days.
Within large states, of an advanced nature, a large fraction of the economic
product is pooled, via taxation, and re-allocated according to democratic decisions. We
have huge firms, in which cooperation is largely decentralized. Trade unions show the
extent of cooperation in firms that is decentralized and tacit when, in labor struggles, they
instruct their members to ‘work to rule.’ In other words, it is wrong to view cooperation
as primarily organized centrally; it’s a false dichotomy to say that competition is
decentralized and cooperation must be centrally planned. By far most instances of
human cooperation are decentralized as well.3
From this perspective, it is quite astonishing that economic theory has hardly
anything to say about cooperation. Our two greatest contributions to understanding
economic activity – the theory of competitive equilibrium, and game theory with its
concomitant concept of Nash equilibrium -- are theories of how agents compete with
each other. Behavior of agents in these theories is autarchic: I decide upon the best
strategy for me under the assumption that others are inert. Indeed, in Walrasian general
equilibrium, a person need not even observe the actions that others are taking: she need
only observe prices, and optimize as an individual, taking prices as given. Nothing like
Tomasello’s joint intentionality exists in these theories: rather, other individuals are
treated as parameters in an agent’s optimization problem.
It would, however, be a mistake to say that economic theory has ignored
cooperation. Informally, lip service is paid to the cooperative tendency of economic
actors: it is commonplace to observe that contracts would not function in a cut-throat
competitive society. There must be trust and convention to grease the wheels of
competition. Nevertheless, this recognition is almost always in the form of the gloss
economists put on their models, not in the guts of the models.
There is, however, one standard theory of cooperation, where cooperative
behavior is enforced as the Nash equilibrium of a game with many stages. There are
typically many Nash equilibria in such games. The ‘cooperative’ one is often identified
as a Pareto efficient equilibrium, where the cooperative behavior is enforced by
punishing players at stage t+1 who failed to play cooperatively at stage t. Since
punishing others is costly to the punisher, those assigned to carry out punishment of
deviants must themselves be threatened with punishment at stage t + 2 , should they fail
to punish. Only if such games have an infinite or indefinite number of stages can this
behavior constitute a Nash equilibrium. For if it were known that the game had only
three stages, then no person in stage 3 will punish deviators from stage 2, because there is
no stage 4 in which they would punished for shirking. So in stage 2, agents will fail to
play cooperatively. By backward induction, the ‘good’ equilibrium unravels. (See
Kandori [1992]. )
What’s interesting about this explanation of cooperation is that it forces
cooperation into the template of non-cooperative Nash equilibrium. I will maintain that4
this is an unappealing solution, and too complex as well. It is a Ptolemaic attempt to use
non-cooperative theory to explain something fundamentally different.
Let me give a simple example, the prisoners’ dilemma, with two players and two
strategies, C(ooperate) and D(efect). In fact, the strategy profile (D,D) is something
stronger than a Nash equilibrium: it’s a dominant strategy equilibrium. If the game is
played with an indefinite number of stages, then the behavior where both players
cooperate at each stage can be sustained as a Nash equilibrium, if punishments are
applied to defectors. I propose, alternatively, that in a symmetric game like this one,
each player may ask himself “What’s the strategy I’d like both of us to play?” This
player is not considering the welfare of the other player: she is asking whether for her
own welfare a strategy profile (C,C) is better than the profile (D,D). The answer is yes,
and if both players behave according to this Kantian protocol (‘take the action I’d like
everyone to take’), then the Pareto efficient solution is achieved in the one-shot game.
What is needed for people to think like this? I believe it is being in a solidaristic
situation. Solidarity is defined as ‘a union of purpose, interests, or sympathies among the
members of a group (American Heritage Dictionary).’ Solidarity, so defined, is not the
action we take together, or the feeling I have towards others, it is a state of the world that
might induce unison action. Solidarity may promote joint action, in the presence of
trust: if I take the action that I’d like all of us to take, I can trust others will take it as well.
To be precise, as we will see, this behavior has good consequences when the game is
symmetric (to be defined below). Symmetry is the mathematical form of ‘a union of
purpose or interests.’ Thus Tomasello’s joint intentionality, for me, is what comes
about when there is a union of a solidaristic state and trust.
Trust, however, must be built up from past experience. I therefore do not claim
that it is rational in a truly one-shot game to ask the Kantian question. Nash equilibrium
is the rational solution of the truly one-shot game. But in real life, we are very often in
situations where trust is warranted, either because of past personal experience with
potential partners, or because of social conventions, of culture. In these situations, trust
exists, and the Kantian question is a natural one to ask.
Now you might respond that, if the game is really embedded in a multi-stage
game of life, then the reason that I take the action I’d like all of us to take is for fear that5
if, instead, I played ‘Defect’ (say, in the prisoners’ dilemma) I will be punished in the
future, or I will fail to find partners to play with me. Indeed, I think some people do
think this way. But many people, I propose, do not. They have embedded the morality
that playing the action I’d like all to play is ‘the right thing to do’ and a person should do
the right thing. This behavior is not motivated by fear of punishment, but by morality.
The morality, however, is not appropriately modeled as an object of preferences, but by a
manner of optimizing. This may seem like a pedantic distinction, but I will argue that it
is not.
Indeed, we now come to the second way that contemporary economics explains
cooperation, and that is under the rubric of behavioral economics. Behavioral
economics has many facets: here I am only concerned with its approach to explaining
cooperation. I claim that the general strategy adopted by behavioral economists to
explain cooperation is to insert exotic arguments into preferences – like a sense of
fairness, a desire for equality, a care for the welfare of others, experiencing a warm glow
– and then to derive the ‘cooperative’ solution as a Nash equilibrium of this new game.
Thus, for example, a player in the prisoners’ dilemma plays C because it would be unfair
to take advantage of an opponent playing C by playing D. In this formulation both (C,C)
and (D,D) would be Nash equilibria, if I incur a psychic cost for playing D against your
C. Or suppose we simply say that the player gets a ‘warm glow’ from playing C (see
Andreoni [1990]). Then the unique Nash equilibrium, if the warm glow is sufficiently
large, will be (C,C).
Indeed, Andreoni’s ‘warm glow’ merits further comment. I think it’s true that
many people get a warm glow from playing the Kantian action, from doing the right
thing. But the warm glow is an unintended side effect, to use Elster’s (1981)
terminology, not the motivation for the action. I teach my daughter the quadratic
formula. She gets it: I enjoy a warm glow. But I did not teach her the formula in order
to generate the warm glow, which came along as a result that I did not intend. Andreoni
has reversed cause and effect. The same criticism applies to explanations of charity.
The Kantian explanation is that I give what I’d like everyone in my situation to give,
rather than my giving because it makes me feel good – which is not to deny that I do feel
good when I do the right thing.6
The Kandori explanation of cooperation as a Nash equilibrium in a multi-stage
game with punishments is what Elster (1989) calls a social norm. To be precise, it is
part of Elster’s characterization of social norms that those who deviate be punished by
others, and those who fail to punish deviators are themselves punished by others.
Doubtless, many examples of cooperation are social norms: but not all are. It has often
been observed by economists that normal preferences for risk will not explain the extent
of tax compliance, given the probabilities of being caught for evading, and the
subsequent (small) fines. In some countries, tax evaders’ names are published in the
newspaper, and there it may well be that compliance is a social norm. In many cities,
large numbers of people recycle their trash. Often, nobody observes whether or not one
recycles. There is no punishment, in these cases, for failing to recycle: but many recycle
nevertheless. Assuming that recycling is somewhat costly, the Nash equilibrium – even
if people value a clean environment – is not to recycle. (I should not recycle if the cost of
recycling to me is greater than the marginal contribution my recycling makes to a clean
environment.) Recycling, I think, is better explained as a Kantian equilibrium. Not
everyone recycles because not everyone thinks Kantian.
People’s trust in others may come with thresholds. I will recycle if I see or read
that fraction q of my community recycles. There is a distribution function of the
thresholds q in the community. In figure 1 such a distribution function is graphed;
there is a stable equilibrium where fraction q * recycle. (There are also unstable
equilibria where fraction 0 or fraction 1 recycle.) I have called these people conditional
Kantians; Elster (2017) calls them quasi-moral (if 1 > q > 0) and reserves the label
‘Kantian’ for those for whom q = 0 . Nash players have q = 1 : they always play Nash,
no matter how many others are playing Kantian.
[place figure 1 here]
There are, I think, three explanations of how workers cooperate when they go on
strike, or why people join revolutionary movements or dangerous demonstrations and
protests against the government. The first, promulgated by Olson (1965), is of the
repeated-game-with-punishments variety. Workers who cross the picket line are beaten7
up. Or, there is a carrot: joining the union comes with side payments. Olson’s
explanation is clearly cooperation-as-a-Nash-equilibrium-with-punishments. Recently,
Barbera and Jackson (2017) explain these actions as occurring because participants enjoy
an expressive value from the action: they value expressing their opposition to the regime
or the boss. This is what I’ve called the behavioral-economics approach: putting exotic
arguments into preferences. I (in press) model strikes as games where players’
strategies are their probabilities of striking: in the case where all preferences are the
same, the Kantian equilibrium is a probability that will maximize my expected income if
everyone strikes with that probability. (With heterogeneous preferences, the story is
more complicated.) Preferences are straight-forward economic preferences, with no
expressive element; nor are there punishments. It is not unusual in this model for the
Kantian equilibrium to be that each strikes with probability one. In reality, we do not
often observe this, because not everyone is a Kantian. There may well be punishments,
in reality, to deter those who would not strike when the strike is on. But it is wrong to
infer that those punishments are the reason that most people strike. The punishments may
be needed only to control a fairly small number of Nash optimizers. And if the workers
are conditional Kantians, or q-Kantians, it is possible for a strike to unravel if even a
small number of Nash players are not deterred from crossing the picket line1.
I have thus far only discussed symmetric games – true solidaristic situations,
.where all payoff functions are the same, up to a permutation of the strategies. I will now
formalize what I have proposed, before going on to the more complicated problem of
games that are not symmetric, where payoff functions are heterogeneous.
12. Simple Kantian equilibrium
Consider a game where all players choose strategies from an interval I of real
numbers, and the payoff function of player i is
V i (E i , E −i ) (2.1)
1
Unravelling will not occur at the q * equilibrium in Figure 1, which is a stable
equilibrium. But it will occur if the equilibrium is at q = 1 .8
where the vector of strategies is denoted E = (E 1 ,..., E n ) and E−i is the vector E without
its ith component.
Definition 2.1 A simple Kantian equilibrium is a vector E* = (E * , E * ,..., E * ) such that:
E * = arg maxV i (E,E,...,E) . (2.2)
E
That is, among all vectors where everyone plays the same strategy2, the strategy that
everyone would choose is E * . This is the formalization of the statement that each plays
the strategy he’d like everyone to play.
Definition 2.2 A game is strictly monotone increasing if each player’s payoff is strictly
monotone increasing in the strategies of the other players. It is strictly monotone
decreasing if it is strictly monotone decreasing in the strategies of the other players.
The standard example of a strictly monotone increasing game is when a person’s
E is her contribution to a public good. The more others contribute, the higher my
welfare. The standard example of monotone decreasing game is the common-pool
resource problem. We all fish on a common lake, and the more others fish, the less
productive is the lake for me.
A symmetric game is one where all agents have the same payoff function, subject
to a permutation of the strategy profile. For my purposes, we may consider symmetric
games where each player’s payoff is a function of her strategy and the sum of all
strategies, that is:
for all i, V i (E i , E −i ) = V *(E i ,E S ) , some V * , (2.3)
where E S ≡ ∑ E i . It is immediate to observe that if a game is symmetric, in the sense
i
of satisfying (2.3), then a simple Kantian equiliabrium exists3.
2
The set of strategy profiles where all players play the same strategy is called an
isopraxis, by J. Silvestre, in his contribution to this issue.
3
Simple Kantian equilibria exist for a broader class of games than symmetric ones (see
Roemer(in press, chapter 2).9
The Nash equilibria of strictly monotone games are Pareto inefficient. The failure
of efficiency of Nash equilibrium in monotone decreasing games is called the tragedy of
the commons, while the failure in monotone increasing games is called the free rider
problem. In the case of monotone increasing games, in Nash equilibrium, people
contribute too little; in the case of monotone decreasing games, they fish too much. But
we have:
Proposition 2.1 The simple Kantian equilibrium of a strictly monotone game (symmetric
or not) is Pareto efficient.
This result is what I referred to earlier when I said that in symmetric game, if
everyone plays the strategy he’d like everyone to play, the result is ‘good.’ Because this
result is so central to the idea of Kantian optimization, I will prove it here.
Proof of Proposition 2.1:
Let the game V = (V 1,...,V n ) be strictly monotone decreasing. Let (E * ,..., E * ) be a
simple Kantian equilibrium. If it is not Pareto efficient, then there is a vector
E = (E 1 ,..., E n ) such that:
(∀i)(V i (E) ≥V i (E *,E *,....,E * )) , (2.4)
where the inequality is strict for at least one index i. Let j be an index such that
E j = min E i . ThenV j (E j ,E j ,...,E j ) >V j (E) . This follows because we have reduced
i
the efforts (E) of some players other than j and increased the efforts of no players,
because E j is minimal among the {E i } . So the strict inequality just stated follows
from the fact that the game is strictly monotone decreasing. But this last inequality
implies that:
V j (E j ,E j ,...,E j ) >V j (E *,E *,...,E * ) , (2.5)
by the application of (2.4). This contradicts the assumption that E* is a simple Kantian
equilibrium, for agent j would prefer that everyone play E j to E * , which proves the
proposition.10
An analogous proof establishes the claim for monotone increasing games. ■
In this sense, simple Kantian equilibrium resolves the inefficiencies due to both
negative and positive externalities that are characteristic of Nash equilibrium4.
The general version of a 2 × 2 symmetric prisoners’ dilemma is given by the
payoff matrix in table 1:
C D
C (0,0) (-c,1)
D (1,-c) (-b,-b)
Table 1. The payoffs are (row player, column player).
where 011
upon the values of b and c. The Nash equilibrium of the game is always (0,0): both
players defect for sure, and the outcome is inefficient.
Let me now make precise one of my criticisms of the behavioral economics. One
can compute that if c < 1 , then the simple Kantian equilibrium of the PD game is that
both players cooperate with probability:
2b + 1− c
p* = , (2.10)
2(1+ b − c)
a probability strictly between zero and one. In other words, full cooperation ( p = 1 ) is
not the simple Kantian equilibrium! Of course, it remains true that ( p*, p*) is Pareto
efficient: in particular, it is better for both players (in ex ante utility) than the strategy
profile (1,1). Now suppose you are a behavioral economist, and you want to insert an
exotic argument into the preferences of the agents so that the Nash equilibrium of the
game will be ( p*, p*) ? How would you do it? Recall that the preferences you create
must not work just for this PD game but for all PD games.
There turns out to be a way, and I believe only one sensible way, of doing this.
Assign each player a new payoff function, which is the sum of the payoff functions of the
two players in the original PD. That is, define:
Q( p,q) = V row ( p,q) + V col ( p,q) . (2.11)
Now consider the game where both players have the payoff function Q – of course, the
row player continues to play p and the column player q. It is not hard to check that the
Nash equilibrium of this game is indeed:
2b + 1− c
p̂ = q̂ = p* = . (2.12)
2(1+ b − c)
So we can rationalize the simple Kantian equilibrium of the PD game as the Nash
equilibrium of a game where each player is maximizing an ‘exotic’ preference order, the
total payoff of the original game.
This might appear to support the behavioral economist’s strategy. Here indeed –
and this can be checked – the simple Kantian equilibrium of a symmetric game is always12
a Nash equilibrium of an altered game where each player’s payoff function is the sum of
all players’ payoff functions in the original game. There are, however, two problems
with this move: first, is it credible to think that’s what players are doing when they play
the cooperative strategy in such a game – that they are attempting to maximize the total
payoff? This is something that can be studied experimentally, and I conjecture it will not
be borne out. But secondly, this trick (of transforming a Kantian equilibrium into a Nash
equilibrium of a game with altered preferences) only works when the game is symmetric.
A bit more on this later.
3. Simple production economies
We now consider a class of simple production economies in which we can study
Kantian optimization in environments with negative and positive externalities. Suppose
there is a production function that produces a desirable consumption good from the
efforts of individuals, according to the production function G(E S ) , where G is strictly
concave, increasing and differentiable, E i is the labor expended by person i, measured
in efficiency units, and E S = ∑ E i . Player i has a utility function u i (x, E) where x is
the consumption good produced. As usual u is concave, increasing in x , decreasing in
E, and differentiable. For the moment, we allow the preferences of individuals to differ.
A. The fishing economy
In the fishing economy, we think of G as production of fish from a lake, which
suffers from congestion externalities, the more labor is expended in fishing: hence, the
strict concavity of G. Recall that an interior allocation is Pareto efficient exactly when :
u2i (x i , E i )
for all i, − i i i
= G ′(E S ) , (3.1)
u1 (x , E )
for this is the statement that the marginal rate of substitution between labor and fish for
each fisher (MRS i ) equals the marginal rate of transformation of labor into fish (MRT).
The allocation rule in this economy is given by:13
Ei
x = S G(E S ) ;
i
(3.2)
E
that is, except for random noise, each fisher receives fish in proportion to the efficiency
units of labor he expends. Another way of putting this is that ‘each fisher keeps her
catch.’
We can define a game: the payoff to fisher i, is given by the function:
Ei
V i (E 1 ,..., E n ) = u i ( S
G(E S ), E i ) . (3.3)
E
It is well-known, that due to the strict concavity of G, the Nash equilibrium of this game
is Pareto inefficient. This is the classical example of the tragedy of the commons. In
Nash optimization, players do not take into account the negative externality they impose
on other fishers by their own fishing. For each additional hour I fish, I lower the
marginal productivity of everyone’s labor on the lake. Indeed, the Nash equilibrium is
the solution of these equations:
u2i E i E i G(E S )
for all i, − = G ′ (E S
) + (1− ) . (3.4)
u1i E S ES ES
In words, the marginal rate of substitution of a fisher is equal to a convex combination of
the marginal rate of transformation and the average product. Only in the case of one
fisher is this allocation Pareto efficient – as long as G is not linear.
To apply the concept of simple Kantian equilibrium, we will assume for the
moment that u i = u for all i. What is the simple Kantian equilibrium of the game? It is
given by solving:
d 1
u( G(nE), E) = 0 , (3.5)
dE n
for the solution of (3.5) is the amount of fishing time that each player would like all to
expend. Taking the derivative, we have:
d 1 1
u( G(nE), E) = u1 G ′(nE)n + u2 = 0
dE n n
(3.6)
u
or G ′(E S ) = − 2 .
u1
But this is the condition for Pareto efficiency! Hence the simple Kantian equilibrium is
Pareto efficient.14
Now, a caveat. You might have observed that the game defined in (3.3) is a
monotone decreasing game, and hence concluded from Prop. 2.1 that the simple Kantian
equilibrium (SKE) is Pareto efficient, and so the derivation (3.6) is redundant. But that
inference is false. What Proposition 2.1 shows is that the SKE is Pareto efficient among
all allocations that can be achieved in the game, so defined : that means among all
allocations in which consumption of fish is proportional to labor expended. But (3.6) is a
much stronger statement: it says that the SKE is Pareto efficient in the set of all feasible
allocations for this economy. There is no way of allocating the total fish caught to the
fishers, through any intricate system of redistribution, that can Pareto dominate the SKE
of the game that models the fishing economy. In other words, we achieve full efficiency
even though we restrict ourselves to allocations where each fisher keeps her catch. The
demonstration in (3.6) is stronger than Proposition 2.1.
Again, I must make the comparison with the behavioral-economics approach.
Because this is a symmetric game (when all utility functions are the same), it is indeed
the case that the Nash equilibrium of the game where each player maximizes the sum of
utilities of all players is the simple Kantian equilibrium given by (3.6). So we have, at
this point, two explanations if a fishing community achieves the Pareto efficient
allocation in which each keeps his catch: either each is fishing the amount of time he
would all everyone to fish, or each is a complete altruist, desiring to maximize total
utility, but optimizing in the manner of Nash.
B. Heterogeneous preferences
We now relax the assumption that all utility functions are the same, and assume
the profile of utility functions is u = (u1,...,u n ) . One simple way to generate
heterogeneous preferences is to say that everyone has the same preferences over fish and
labor time, but since fishers have different skill levels, this induces heterogeneous
preferences over fish and efficiency units of labor. Recall that the relevant labor in our
models is the latter.
It’s now the case that simple Kantian equilibria will generally not exist: the labor
that I would most like everyone to expend is the different from the labor you would most15
like everyone to expend. There is, however, a generalization of Kantian optimization
that we can use with heterogeneous preferences.
Suppose at an allocation E = (E 1 ,..., E n ) I am contemplating increasing my
fishing time by 5%. I ask myself: How would I like it if everyone increased her fishing
time by 5%? The implicit moral commandment here is that I should only increase my
fishing time by 5% if I’d be happy were everyone to do the same. It’s a way of making
me take into account the negative externality created by my contemplated action. But I
don’t ask myself how would my increased fishing would affect others, but rather how, if
they emulated my action, their increased fishing times would affect me. Don’t worry
too much at this point about the psychological realism of this question. Instead, let’s
study the properties of an equilibrium when everyone thinks in this way.
Definition 3.1 An allocation of efforts E = (E 1 ,..., E n ) is a multiplicative Kantian
equilibrium (a K × equilibrium) if nobody would like to rescale everybody’s fishing time
by any non-negative scale factor. That is:
⎛ rE i ⎞
(∀i) ⎜ arg max u i ( S G(rE S ),rE i ) = 1⎟ (3.7)
⎝ r≥0 rE ⎠
I have stated the definition for the fishing economy, but in the attached footnote, I state it
for a general game in normal form.5
We have the general result, using the definition in the footnote:
Proposition 3.1 If E = (E 1 ,..., E n ) is a positive K × equilibrium of a strictly monotone
game {V i } , then it is Pareto efficient in the game.
The proof of this proposition is very similar to the proof of proposition 2.1.
In particular, the fishing game of (3.7) is a strictly monotone decreasing game.
So Proposition 3.1 implies that the multiplicative Kantian equilibrium, if it exists, is
5
Let the payoff functions of the game be {V i } where the strategy space for each player
is an interval of real numbers. Then (E 1 ,..., E n ) is a K × equilibrium of the game when
(∀i)(arg maxV i (rE) = 1) .
r ≥016
Pareto efficient in the game. As before, we must recall that being efficient ‘in the game’
means in the class of allocations where each keeps his catch. But we could ask for a
stronger result: is the K × equilibrium efficient in the economy?
We have:
Proposition 3.2 Any strictly positive K × equilibrium of the fishing game is Pareto
efficient in the economy.
Again, because this is a key result, let’s prove it.
Proof:
1. By concavity of the utility functions and G, it is only necessary to show that :
⎛ d Ei ⎞
(∀i) ⎜ |r=1 u i ( S G(rE S ),rE i = 0 ⎟ ⇒ the allocation is Pareto efficient.
⎝ dr E ⎠
This suffices, because the antecedent to the implication sign is the first-order
characterization of K × equilibrium.
2. Compute that:
d Ei i E
i
ui ( S
G(rE S
),rE i
) = u1 ⋅ S
G ′(E S )E S + u2i E i = 0 . (3.8)
dr r=1 E E
Using the assumption that E i > 0 , this equation reduces to:
⎛ u2i ⎞
(∀i) ⎜ − i = G ′(E S )⎟ , (3.9)
⎝ u 1 ⎠
which is exactly the condition for Pareto efficiency at an interior solution. ■
As with the case of homogeneous preferences, we can therefore assert a stronger
statement than Proposition 3.1: in the fishing economy, multiplicative Kantian
optimization resolves the tragedy of the commons that afflicts Nash equilibrium. (Recall
the Nash equilibrium of the fishing game is given by (3.4).)
Do such equilibria exist? These are allocations in which fish consumed is
proportional to labor expended and the allocation is Pareto efficient. The idea of looking
for such allocations is due to Joaquim Silvestre. In Roemer and Silvestre (1993), we17
proved that in economies much more general than the simple production economies
studied here, such allocations always exist. Silvestre’s question was asked in the context
of thinking about the socialist ideal: an allocation in which the total product is distributed
in proportion to labor expended, and is Pareto efficient. Remarkably, perhaps, this
question had not been raised earlier by mathematical socialist economists like Oscar
Lange and Michio Morishima. It was not, however, until several years later that I
observed that these allocations, which Silvestre and I called proportional solutions, had
the property of being stable with respect to the kind of optimization that I now call
Kantian (see Roemer [1996, Theorem 6.6]). Also, as an illustration of ‘thinking slow,’
the much simpler idea of simple Kantian equilibrium in symmetric games did not occur to
me for another twenty years.
We can now repeat our earlier question: Is there a way of altering preferences in
the fishing game so that the Nash equilibrium of the altered game is the Pareto efficient
allocation in which each keeps his catch (i.e., the multiplicative Kantian equilibrium)?
(There is often a unique such allocation; in any case, there is a finite number of such
allocations.) The answer is there is no simple way of doing this: the idea of giving all
players the desire to maximize total utility no longer works. For further discussion of
the representation of Kantian equilibria as Nash equilibria of games with exotic
preferencers, I refer the reader to Roemer (in press, chapter 6).
In other words, in games that are fairly complicated, the behavioral economist’s
strategy of altering preferences in order to achieve the good (cooperative) solution as the
Nash equilibrium of a game does not work. Essentially, one has to know what the good
solution is a priori and then one can jimmy preferences to give the right answer under
Nash optimization. But this procedure cannot be regarded as decentralizing cooperation.
In contrast, multiplicative Kantian optimization decentralizes cooperation, in the exact
sense that Nash optimization decentralizes competition. As with Nash, there is no
obvious solution to the dynamic problem of how one converges to a Nash (or Kantian)
equilibrium. But, as with Nash, both kinds of equilibria are stable given the kind of
optimization that players are using, once equilibrium is achieved.18
I will have more to say about the realism of proposing that players in a game
might optimize in the multiplicative Kantian fashion – but later. At this point, however,
I still want to explicate the properties of this kind of thinking.
C. The hunting economy
Consider the game that according to anthropologists characterized early hunting
societies. Unlike fishers, hunters divided the catch equally (more or less). A group of
hunters goes out into the bush for a few days. They return with their catch, and divide it
equally among the group.
Here the allocation rule is:
G(E S )
x =
i
, (3.10)
n
which is the equal-division rule. The game defined by the equal-division rule is given
by:
G(E S ) i
V ED,i (E 1 ,..., E n ) = u i ( ,E ) . (3.11)
n
This is a game with positive externalities: the total catch is a public good. The game is
strictly monotone increasing. Again, the Nash equilibrium, which is given by the next
set of equations, is Pareto inefficient as long as there is more than one hunter:
u2i G ′(E S )
− = . (3.12)
u1i n
Why is the Nash equilibrium inefficient? Because a hunter might like to take a
nap for several hours under a bush, given what others are doing. He contributes less to
the public good than efficiency requires. There is a free rider problem.
It turns out that multiplicative Kantian optimization does not resolve this
inefficiency. But there is a kind of Kantian optimization that does. Suppose the hunter
asks himself, “I’d like to take a two hour nap. But how would I feel if everyone took at
two hour nap?” If the moral commandment is to take the nap only if the answer to this
question is affirmative, then an equilibrium is defined as follows:19
Definition 3.2 An allocation E = (E 1 ,..., E n ) is an additive Kantian equilibrium (K + ) if
and only if:
⎛ G(E S + nr) i ⎞
(∀i) ⎜ arg max u i ( , E + r) = 0 ⎟ . (3.13)
⎝ r n ⎠
In other words, nobody would like to translate the effort vector by any number (positive
or negative)6.
I will now leave it to the reader to verify:
Proposition 3.3 Any additive Kantian equilibrium of the hunting game is Pareto efficient
in the economy.
Thus, additive Kantian optimization resolves the under-provision of the public
good in the hunting economy.
Kantian optimization always takes the form of each asking: “If we all take a
similar symmetric action, what would I like that action to be?” In the case when the
game is symmetric, the action is ‘making the same contribution.’ In the fishing game, the
action is ‘rescaling the current allocation by a common factor;’ in the hunting game, it is
‘translating the current allocation by a common factor.’
D. More general allocation rules in simple production economies
Let’s define an allocation rule for an economic environment specified by the data
(u1 ,...,u n ,G) as the share of output θi each individual receives as a function of the
vector of effort contributions. We have studied two rules, the proportional and equal-
division rules, whose share rules are:
Ei 1
θ Pr,i (E 1 ,..., E n ) = S
, θ ED,i (E 1 ,..., E n ) = for all i . (3.14)
E n
6
For general games, (E 1 ,..., E n ) is an additive Kantian equilibrium if:
(∀i) ⎛ 0 = arg max V i (E 1 + r,..., E n + r)⎞ .
⎝ r ⎠20
Indeed, these are surely the two classical notions of fair division of a jointly produced
output. We have shown that each of these rules can be efficiently implemented -- more
strongly, efficiently decentralized—by a specific kind of Kantian optimization.
We can also think more generally about kinds of Kantian symmetrical treatment.
Define a Kantian variation as a function ϕ(E,r) where E is a contribution and r is a
number where the following holds:
ϕ(E,1) ≡ 1, ϕ(E,r) is increasing in r . (3.15)
The multiplicative Kantian variation is given by:
ϕ × (E,r) = rE (3.16)
and the additive Kantian variation is given by:
ϕ + (E,r) = E + r − 1 . (3.17)
Now we say that the allocation rule θ is efficiently implemented by the Kantian
variation ϕ on the set of economies {(u1 ,...,u n ,G)} if for all economies:
(∀i) ⎛ 1 = arg max u i (θi (ϕ(E,r)G(∑ ϕ(E i ,r)),ϕ(E i ,r))⎞ , (3.18)
⎝ r ⎠
where ϕ(E,r) ≡ (ϕ(E 1 ,r),...,ϕ(E n ,r)) .
Equations (3.18) simply state the generalization of multiplicative and additive
Kantian equilibrium to other allocation rules, and other types of Kantian variation. They
say that an allocation has the property that, for the given allocation rule θ , nobody would
like to vary the entire effort vector according to any transformation permitted by the
variation defined by ϕ .
Definition 3.2 The pair (θ,ϕ) is an efficient Kantian pair if the allocation rule θ is
efficiently implemented as a K ϕ equilibrium on all economies of the form (u1 ,...,u n ,G) .
What we have thus far shown is that (θPr ,ϕ × ) and (θ ED ,ϕ + ) are efficient Kantian
pairs. The question is: Are there any others?21
The answer is that there is a unidimensional continuum of such pairs, of which the
two we have studied are the endpoints. Consider allocation rule of the form:
Ei + β
θβ,i (E 1 ,..., E n ) = (3.19)
E S + nβ
where 0 ≤ β < ∞ . Notice that θ0 is the proportional rule, and that as β
approaches infinity, θβ approaches the equal-division rule. Because of this, let’s define
1
θ∞ = θ ED = . We have:
n
Proposition 3.4 For every rule θβ , there is a Kantian variation that implements it
efficiently on the domain of production economies we are considering. Furthermore,
there are no other allocation rules that can be efficiently implemented on this domain
with respect to any Kantian variation7.
(For proof, see Roemer (in press, Corollary 4.4).)
What do the rules θβ look like? It is easy to show that they are ‘convex
combinations’ of the proportional rule and equal division rule. Let’s solve the following
equation for λ :
Ei + β Ei 1
= λ + (1− λ) . (3.20)
E + nβ
S
E S
n
ES
The solution is λ = S . The important fact is that the value of λ is independent
E + nβ
of i. This means that the rule θβ is simply this: it takes a share λ of the output G(E S )
and distributes it to the participants in proportion to the efficiency units of labor
expended, and it distributes the rest of the output equally to all. The share λ , however,
7
There are two caveats. If β < 0 , the allocation rules can also be efficiently
implemented; however, Kantian allocations may not exist. And for β = 0 we must
insist that the allocation be strictly positive. (The zero vector is a multiplicative Kantian
equilibrium but it is not Pareto efficient.)22
depends upon the equilibrium. We know that as β travels from zero to infinity, the
share λ travels from one to zero: but that is all that we can say.
It’s for this reason that I put ‘convex combination’ in quotations marks earlier.
We cannot specify the share λ to be – say – one-half, and then quickly choose the right
allocation rule to implement that share. To say this mathematically, the mapping from
λ to β is complicated, as to know it we have to compute the equilibrium to know E S .
Only after knowing the equilibrium for each β can we find the one that implements a
particular combination of proportional and equal division.
The upshot of this discussion is that the class of allocation rules that can be
efficiently implemented by any kind of Kantian thinking is precisely the class of rules
generated by the two classical egalitarian methods of distribution: in proportion to effort
and equally. As these rules have a venerable history as rules of fair allocation,
Proposition 3.4 re-enforces the view that Kantian optimization is the right way of
thinking about morality and, may we say, cooperation.
I conclude this section with a remark about the behavioral feasibility of Kantian
optimization. I think simple Kantian optimization is natural – there are many symmetric
games that describe our social interactions, and in those games, I think many people ask
the Kantian question: what’s the action I’d like everyone to take? Multiplicative and
additive Kantian optimization are, however, not natural. I think symmetry plays a large
role in our conceptions of fairness, and the symmetrical treatment of all characteristic of
Kantian optimization recommends it as a way of implementing fairness. But I do not
claim societies have discovered these complex ways of optimizing. Rather, I see the
approach as prescriptive. If the fairness involved in these kinds of Kantian optimization
appeals to people, we may recommend –- to a fishing community, for example – the
multiplicative Kantian equilibrium as a solution to their problem. Doing so, of course,
would require the planner to know the preferences of the fishers, unless some process I
do not yet understand could lead dynamically to the Kantian equilibrium of the game.
I have given examples where Kantian optimization resolves a tragedy of the
commons, and a free rider problem. In fact, we have two general results: first, ‘any’
kind of Kantian optimization results in Pareto efficient equilibria in all strictly monotone
games, and second, generically, the Nash equilibria in strictly monotone games are Pareto23
inefficient. (See Roemer (in press, chapter 2.) Of course, the value of these statements
depends upon the existence of the equilibria in question.
Thus, cooperation in the sense of Kantian reasoning resolves, generally, the two
major failures of Nash optimization.
4. Market socialism
Thus far, I have described economies that do not trade on markets. Indeed, there
is nothing that can be thought of as comprising trade in the fishing and hunting
economies. We now ask what role Kantian optimization can play in market economies.
Indeed, I will propose that Kantian optimization can play an important role in a
design for market socialism. The ‘design problem’ for socialism is stated by the
philosopher G.A. Cohen as follows:
In my view, the principal problem that faces the socialist ideal is that we do not
know how to design the machinery that would make it run. Our problem is not,
primarily, human selfishness, but our lack of a suitable organizational technology: our
problem is a problem of design. It may be an insoluble design problem, and it is a
design problem that is undoubtedly exacerbated by our selfish propensities, but a
design problem, so I think, is what we’ve got. (Cohen, 2009)
What Cohen means is this. Capitalism has a design consisting of private property rights
in labor and productive assets, and free trade, mediated by prices that equilibrate supply
and demand. The ethos that makes capitalism work is the maximization of self-regarding
preferences in an autarchic manner: the individualist ethos is captured not only in the
nature of preferences, but in the manner in which people optimize ( à la Nash). Cohen
also views the motivation for economic activity under capitalism as being ‘greed and
fear,’ a point with which I do not completely agree.
Socialism, however, is supposed to be an economic system characterized by
cooperation. The natural question becomes, how can one design an economic system
based on cooperation to deliver good results? The first theorem of welfare economics
for capitalist economies is the main formalized example of capitalism’s good result. Can24
we design a mechanism, that given a socialist ethos, would deliver good results? A
good result for socialism should demand a higher standard than for capitalism: we desire
to have not only efficiency but also a substantial degree of income equality.
Cohen, in the above quotation, says that it is not primarily (selfish) human
nature that poses a problem for socialism, it is the lack of a solution to this design
problem. Cohen, evidently, is optimistic about human nature – and there is indeed good
reason to be so. Many people, in today’s capitalist societies, do not plan their lives to
maximize their wealth, but to do work that is useful for society. I agree with Cohen that
human nature is not the primary obstacle to socialism, but the lack of a design that can
decentralize economic activity to achieve good results, given that a socialist ethos exists
in the population.
To state the problem slightly differently, recall the huge effect that Marx’s theory
of historical materialism had on engendering an international socialist movement. Marx
offered – if not a design in the sense here being discussed – a vision of the feasibility and
indeed historical necessity of socialism. We need not here debate the validity of that
vision: what’s salient is that, possessing this vision, millions of people organized to
attempt to realize it. Now that vision has soured, due to the experiences of twentieth
century socialism, constrained as they were by political authoritarianism and the fear of
introducing markets. Having a design for how socialism could work, given a willing
population, is of primary importance in rekindling the socialist vision.
I am less suspicious of markets than Cohen was, and so I will propose how
incorporating cooperative optimization into a market economy can produce eqiuilibria
which are decentralized, efficient, and equitable.
The model is considerably more general than the one I now exposit, for purposes
of clarity. Assume there is one produced good, which is used both for investment and
consumption. The good is produced from capital and labor, according to a production
function G(K, L) , which is operated by a firm. The firm is owned by private citizens
and the state. The share in the firm’s profits of private citizen i is θi , for i = 1,....,n,
n
and the state’s share is θ0 : so ∑θ i
= 1 . Individuals, as well as owning shares of the
i=0
firm, have endowments of efficiency units of labor in the amount ω i . Citizen i has a25
preference order over consumption of the good and labor expended, denoted in efficiency
units, represented by a utility function u i (x, E) . The state is endowed with an amount
of capital K 0 -- this endowment was presumably produced in the previous period, not
formalized in the model. Individuals own no capital, nor do they hold inventories of the
consumption good. (The capital good and consumption good are, as I said, the same
good, except for their vintage.)
So far, I have described exactly the set-up of an Arrow-Debreu economy, except
for the state’s partial ownership of the firm. There will be three prices in this economy, a
price for output p, an interest rate for borrowing capital r, and a wage w per unit of
efficiency labor. (Because the capital with which the state is endowed was produced in
the past, its rental price will differ from the price of the good produced in the present
period.) The firm behaves exactly as an Arrow-Debreu firm: it demands capital and
labor and supplies the good in order to maximize profits. All incomes in the economy,
except the state’s revenues, will be taxed at an exogenous rate t ∈[0,1] , and the
revenues will returned to the citizenry as a demogrant: that is, divided equally among
them. This is a flat, or affine, income tax.
How does the state behave? It supplies capital to the firm to maximize its
income, which it uses to purchase the good produced by firm, to be used for next period’s
investment. In fact, the model is truncated: there is only one period, for purposes of
simplicity. So the firm demands capital in the present period, which is uniquely supplied
by the state, and supplies the state with the good produced at present, which the state
purchases with its revenues (rents and capital income).
How do consumer-workers behave? Given a worker’s labor supply E i to the
firm, he uses his revenues to purchase the good for consumption. Workers’ revenues
come from three sources: their after-tax wage and profit income, and the demogrant.
The only substantial way in which the model differs from the standard Arrow-
Debreu model, besides the state’s role, is in the determination of the labor supplies of
workers. The vector of labor supplies (E 1 ,..., E n ) must be an additive Kantian
equilibrium of a game to be defined below.26
Finally, a price vector ( p,w,r) comprises a Walras-Kant equilibrium at the tax
rate t if, given the optimizing behaviors described, all markets clear: that is, the supply
of the good by the firm equals the demand of consumers for the good plus the demand by
the state for the good (investment), the supply of capital by the state equals the firm’s
demand for capital, and the total efficiency units of labor supplied by workers equals the
demand for labor by the firm.
We state the definition formally.
Definition 4.1 A Walras-Kant market-socialist equilibrium at tax rate t, consists of:
i. a price vector (p,w,r) ,
ii. labor and capital demands by the firm of D and K , respectively,
iii. labor supplies E i by all workers i = 1,...,n to the firm,
iv. for all private agents i, commodity demands x i for the good, and a demand for
the good by the state of x 0 ,
such that:
v. at given prices, (K,D) maximizes profits of the firm;
vi. the labor supply vector E = (E 1 ,..., E n ) constitutes an additive Kantian
equilibrium at the given prices of the game V+ defined in equation (4.4) below;
vii. x i maximizes the utility of agent i, given prices, her labor supply, and her
income (that is, its purchase exhausts her budget);
x 0 maximizes the state’s utility u subject to its budget constraint
0
viii.
px 0 ≤ θ0Π + rK 0 , and
ix. all markets clear; that is, D = E S , x S = G(K,D) , and K 0 = K .
We must now be more precise in order to define the game of which the effort
vector must be an additive Kantian equilibrium. First, define the income of the state,
which will be:
I 0 ( p,r,w, E S ) = θ0Π(K 0 , E S ) + rK 0 , (4.1)
where the firm’s profits are:27
Π(K 0 , E S ) = pG(K 0 , E S ) − wE S − rK 0 . (4.2)
In words, (4.1) states that the state’s income consists of its share of firm profits plus the
interest on its investment. The income of worker i is given by:
t
I i (E 1 ,..., E n ) = (1− t)wE i + (1− t)θi Π(K 0 , E S ) + ( pG(K 0 , E S ) − I 0 (K 0 , E S )) . (4.3)
n
The first term on the right-hand side of (4.3) is agent i’s after-tax wage income, the
second term is her after-tax profit income, and the third term is the demogrant, which is
the agent’s per capita share of tax revenues, where taxes are levied on all incomes except
the state’s.
The payoff function for worker i is given by :
⎛ I i (E 1 ,...,E n ) i ⎞
V (E ,E ,...,E ) = u ⎜
+
i 1 2 n i
,E ⎟ . (4.4)
⎝ p ⎠
That is, the worker’s utility is generated by spending her entire income on the
consumption good, and her labor. This completes the definition of equilibrium.
If we assume that the depreciation rate of capital is zero, then the state’s capital,
I 0 (K 0 , E S )
to be used in the ‘next period,’ will be K 0 + .
p
I emphasize that the only substantial ways in which this equilibrium differs from a
private-ownership Arrow-Debreu equilibrium are that the state carries out all the
investment, and the labor-supply decisions do not comprise a Nash equilibrium for
workers, but an additive Kantian equilibrium. In determining their supplies of labor,
workers do not ask whether supplying an extra day’s labor generates an after-tax wage
that compensates for the extra disutility of labor, assuming all other workers’ supplies of
labor remain fixed, but rather, whether the extra day’s income compensates for the extra
day’s work, assuming all others expend an extra day’s labor as well, which would
increase substantially the value of the demogrant.
The consequence of this single change in optimizing behavior is the following:You can also read
